Computing free energy is a fundamental problem in statistical physics.Recently,two distinct methods have been developed and have demonstrated remarkable success:the tensor-network-based contraction method and the neur...Computing free energy is a fundamental problem in statistical physics.Recently,two distinct methods have been developed and have demonstrated remarkable success:the tensor-network-based contraction method and the neural-network-based variational method.Tensor networks are accurate,but their application is often limited to low-dimensional systems due to the high computational complexity in high-dimensional systems.The neural network method applies to systems with general topology.However,as a variational method,it is not as accurate as tensor networks.In this work,we propose an integrated approach,tensor-network-based variational autoregressive networks(TNVAN),that leverages the strengths of both tensor networks and neural networks:combining the variational autoregressive neural network’s ability to compute an upper bound on free energy and perform unbiased sampling from the variational distribution with the tensor network’s power to accurately compute the partition function for small sub-systems,resulting in a robust method for precisely estimating free energy.To evaluate the proposed approach,we conducted numerical experiments on spin glass systems with various topologies,including two-dimensional lattices,fully connected graphs,and random graphs.Our numerical results demonstrate the superior accuracy of our method compared to existing approaches.In particular,it effectively handles systems with longrange interactions and leverages GPU efficiency without requiring singular value decomposition,indicating great potential in tackling statistical mechanics problems and simulating high-dimensional complex systems through both tensor networks and neural networks.展开更多
Constraint satisfaction problems(CSPs)are a class of problems that are ubiquitous in science and engineering.They feature a collection of constraints specified over subsets of variables.A CSP can be solved either dire...Constraint satisfaction problems(CSPs)are a class of problems that are ubiquitous in science and engineering.They feature a collection of constraints specified over subsets of variables.A CSP can be solved either directly or by reducing it to other problems.This paper introduces the Julia ecosystem for solving and analyzing CSPs with a focus on the programming practices.We introduce some important CSPs and show how these problems are reduced to each other.We also show how to transform CSPs into tensor networks,how to optimize the tensor network contraction orders,and how to extract the solution space properties by contracting the tensor networks with generic element types.Examples are given,which include computing the entropy constant,analyzing the overlap gap property,and the reduction between CSPs.展开更多
We propose a generalized Lanczos method to generate the many-body basis states of quantum lattice models using tensor-network states （TNS）. The ground-state wave function is represented as a linear superposition com...We propose a generalized Lanczos method to generate the many-body basis states of quantum lattice models using tensor-network states （TNS）. The ground-state wave function is represented as a linear superposition composed from a set of TNS generated by Lanczos iteration. This method improves significantly the accuracy of the tensor-network algorithm and provides an effective way to enlarge the maximal bond dimension of TNS. The ground state such obtained contains significantly more entanglement than each individual TNS, reproducing correctly the logarithmic size dependence of the entanglement entropy in a critical system. The method can be generalized to non-Hamiltonian systems and to the calculation of low-lying excited states, dynamical correlation functions, and other physical properties of strongly correlated systems.展开更多
Tensor networks are used to describe the ground state wavefunction of the quantum many-body system.Recently,it has been shown that a tensor network can generate the anti-de Sitter(AdS)geometry by using the entanglemen...Tensor networks are used to describe the ground state wavefunction of the quantum many-body system.Recently,it has been shown that a tensor network can generate the anti-de Sitter(AdS)geometry by using the entanglement renormalization approach,which provides a new way to realize bulk reconstruction in the AdS/conformal field theory correspondence.However,whether the dynamical connections can be found between the tensor network and gravity is an important unsolved problem.In this paper,we give a novel proposal to integrate ideas from tensor networks,entanglement entropy,canonical quantization of quantum gravity and the holographic principle and argue that the gravitational dynamics can be generated from a tensor network if the wave function of the latter satisfies the Wheeler–DeWitt equation.展开更多
The Ryu-Takayanagi(RT)formula plays a large role in the current theory of gauge-gravity duality and emergent geometry phenomena.The recent reinterpretation of this formula in terms of a set of"bit threads"is...The Ryu-Takayanagi(RT)formula plays a large role in the current theory of gauge-gravity duality and emergent geometry phenomena.The recent reinterpretation of this formula in terms of a set of"bit threads"is an interesting effort in understanding holography.In this study,we investigate a quantum generalization of the"bit threads"based on a tensor network,with particular focus on the multi-scale entanglement renormalization ansatz(MERA).We demonstrate that,in the large c limit,isometries of the MERA can be regarded as"sources"(or"sinks")of the information flow,which extensively modifies the original picture of bit threads by introducing a new variableρ:density of the isometries.In this modified picture of information flow,the isometries can be viewed as generators of the flow.The strong subadditivity and related properties of the entanglement entropy are also obtained in this new picture.The large c limit implies that classical gravity can emerge from the information flow.展开更多
This paper investigates the networked evolutionary model based on snow-drift game with the strategy of rewards and penalty. Firstly, by using the semi-tensor product of matrices approach, the mathematical model of the...This paper investigates the networked evolutionary model based on snow-drift game with the strategy of rewards and penalty. Firstly, by using the semi-tensor product of matrices approach, the mathematical model of the networked evolutionary game is built. Secondly, combined with the matrix expression of logic, the mathematical model is expressed as a dynamic logical system and next converted into its evolutionary dynamic algebraic form. Thirdly, the dynamic evolution process is analyzed and the final level of cooperation is discussed. Finally, the effects of the changes in the rewarding and penalty factors on the level of cooperation in the model are studied separately, and the conclusions are verified by examples.展开更多
This article discusses the covariance correlation tensor (CCT) in quantum network theory for four Bell bases in detail. Furthermore, it gives the expression of the density operator in terms of CCT for a quantum networ...This article discusses the covariance correlation tensor (CCT) in quantum network theory for four Bell bases in detail. Furthermore, it gives the expression of the density operator in terms of CCT for a quantum network of three nodes, thus gives the criterion of entanglement for this case, i.e. the conditions of complete separability and partial separability for a given quantum state of three bodies. Finally it discusses the general case for the quantum network of nodes.展开更多
目的探讨抑郁障碍和双相障碍患者脑白质网络节点强度的差异,分析患者不同脑区的结构连接受损情况及其在鉴别中的作用。方法纳入91例基线诊断为抑郁发作的患者,经过≥9年的自然观察随访后,最终确定23例维持抑郁障碍诊断(单相组)和18例维...目的探讨抑郁障碍和双相障碍患者脑白质网络节点强度的差异,分析患者不同脑区的结构连接受损情况及其在鉴别中的作用。方法纳入91例基线诊断为抑郁发作的患者,经过≥9年的自然观察随访后,最终确定23例维持抑郁障碍诊断(单相组)和18例维持双相障碍诊断(双相组)的患者纳入分析。同时纳入30名健康对照者(对照组)。受试者在入组时均接受弥散张量成像扫描,采用确定性纤维追踪技术构建脑白质结构加权网络。比较三组间脑白质网络的节点连接强度差异,进一步采用受试者操作特征(receiver operator characteristic,ROC)曲线评估差异脑区对抑郁障碍和双相障碍鉴别诊断的价值。结果双相组在左前扣带回的节点强度较单相组降低(3.89±0.76 vs.4.74±0.60),在右尾状核(4.94±1.26 vs.3.46±0.99)、右苍白球(1.98±0.67 vs.1.25±0.29)的节点强度较单相组升高(P<0.01,FWE校正)。左前扣带回、右尾状核、右苍白球3个脑区的连接强度联合鉴别抑郁障碍和双相障碍绘制ROC曲线,曲线下面积(area under the curve,AUC)为0.95(95%CI:0.91~0.99;P<0.01),敏感度0.89,特异度0.87。结论脑结构网络的节点强度差异可以作为一个潜在的影像学生物标志物识别抑郁障碍和双相障碍,联合差异脑区的节点强度可以得到更好的识别率。展开更多
基金supported by Projects 12325501,12047503,and 12247104 of the National Natural Science Foundation of ChinaProject ZDRW-XX-2022-3-02 of the Chinese Academy of Sciencessupported by the Innovation Program for Quantum Science and Technology project 2021ZD0301900。
文摘Computing free energy is a fundamental problem in statistical physics.Recently,two distinct methods have been developed and have demonstrated remarkable success:the tensor-network-based contraction method and the neural-network-based variational method.Tensor networks are accurate,but their application is often limited to low-dimensional systems due to the high computational complexity in high-dimensional systems.The neural network method applies to systems with general topology.However,as a variational method,it is not as accurate as tensor networks.In this work,we propose an integrated approach,tensor-network-based variational autoregressive networks(TNVAN),that leverages the strengths of both tensor networks and neural networks:combining the variational autoregressive neural network’s ability to compute an upper bound on free energy and perform unbiased sampling from the variational distribution with the tensor network’s power to accurately compute the partition function for small sub-systems,resulting in a robust method for precisely estimating free energy.To evaluate the proposed approach,we conducted numerical experiments on spin glass systems with various topologies,including two-dimensional lattices,fully connected graphs,and random graphs.Our numerical results demonstrate the superior accuracy of our method compared to existing approaches.In particular,it effectively handles systems with longrange interactions and leverages GPU efficiency without requiring singular value decomposition,indicating great potential in tackling statistical mechanics problems and simulating high-dimensional complex systems through both tensor networks and neural networks.
基金funded by the National Key R&D Program of China(Grant No.2024YFE0102500)the National Natural Science Foundation of China(Grant No.12404568)+1 种基金the Guangzhou Municipal Science and Technology Project(Grant No.2023A03J00904)the Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area,China and the Undergraduate Research Project from HKUST(Guangzhou).
文摘Constraint satisfaction problems(CSPs)are a class of problems that are ubiquitous in science and engineering.They feature a collection of constraints specified over subsets of variables.A CSP can be solved either directly or by reducing it to other problems.This paper introduces the Julia ecosystem for solving and analyzing CSPs with a focus on the programming practices.We introduce some important CSPs and show how these problems are reduced to each other.We also show how to transform CSPs into tensor networks,how to optimize the tensor network contraction orders,and how to extract the solution space properties by contracting the tensor networks with generic element types.Examples are given,which include computing the entropy constant,analyzing the overlap gap property,and the reduction between CSPs.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11190024 and 11474331)
文摘We propose a generalized Lanczos method to generate the many-body basis states of quantum lattice models using tensor-network states （TNS）. The ground-state wave function is represented as a linear superposition composed from a set of TNS generated by Lanczos iteration. This method improves significantly the accuracy of the tensor-network algorithm and provides an effective way to enlarge the maximal bond dimension of TNS. The ground state such obtained contains significantly more entanglement than each individual TNS, reproducing correctly the logarithmic size dependence of the entanglement entropy in a critical system. The method can be generalized to non-Hamiltonian systems and to the calculation of low-lying excited states, dynamical correlation functions, and other physical properties of strongly correlated systems.
基金supported by the National Natural Science Foundation of China under Grant No.11675272supported by China Postdoctoral Science Foundation(No.2019M653137)。
文摘Tensor networks are used to describe the ground state wavefunction of the quantum many-body system.Recently,it has been shown that a tensor network can generate the anti-de Sitter(AdS)geometry by using the entanglement renormalization approach,which provides a new way to realize bulk reconstruction in the AdS/conformal field theory correspondence.However,whether the dynamical connections can be found between the tensor network and gravity is an important unsolved problem.In this paper,we give a novel proposal to integrate ideas from tensor networks,entanglement entropy,canonical quantization of quantum gravity and the holographic principle and argue that the gravitational dynamics can be generated from a tensor network if the wave function of the latter satisfies the Wheeler–DeWitt equation.
基金Supported in part by the National Natural Science Foundation of China(11975116,11665016,11563006)Jiangxi Science Foundation for Distinguished Young Scientists(20192BCB23007)。
文摘The Ryu-Takayanagi(RT)formula plays a large role in the current theory of gauge-gravity duality and emergent geometry phenomena.The recent reinterpretation of this formula in terms of a set of"bit threads"is an interesting effort in understanding holography.In this study,we investigate a quantum generalization of the"bit threads"based on a tensor network,with particular focus on the multi-scale entanglement renormalization ansatz(MERA).We demonstrate that,in the large c limit,isometries of the MERA can be regarded as"sources"(or"sinks")of the information flow,which extensively modifies the original picture of bit threads by introducing a new variableρ:density of the isometries.In this modified picture of information flow,the isometries can be viewed as generators of the flow.The strong subadditivity and related properties of the entanglement entropy are also obtained in this new picture.The large c limit implies that classical gravity can emerge from the information flow.
文摘This paper investigates the networked evolutionary model based on snow-drift game with the strategy of rewards and penalty. Firstly, by using the semi-tensor product of matrices approach, the mathematical model of the networked evolutionary game is built. Secondly, combined with the matrix expression of logic, the mathematical model is expressed as a dynamic logical system and next converted into its evolutionary dynamic algebraic form. Thirdly, the dynamic evolution process is analyzed and the final level of cooperation is discussed. Finally, the effects of the changes in the rewarding and penalty factors on the level of cooperation in the model are studied separately, and the conclusions are verified by examples.
文摘This article discusses the covariance correlation tensor (CCT) in quantum network theory for four Bell bases in detail. Furthermore, it gives the expression of the density operator in terms of CCT for a quantum network of three nodes, thus gives the criterion of entanglement for this case, i.e. the conditions of complete separability and partial separability for a given quantum state of three bodies. Finally it discusses the general case for the quantum network of nodes.
文摘目的探讨抑郁障碍和双相障碍患者脑白质网络节点强度的差异,分析患者不同脑区的结构连接受损情况及其在鉴别中的作用。方法纳入91例基线诊断为抑郁发作的患者,经过≥9年的自然观察随访后,最终确定23例维持抑郁障碍诊断(单相组)和18例维持双相障碍诊断(双相组)的患者纳入分析。同时纳入30名健康对照者(对照组)。受试者在入组时均接受弥散张量成像扫描,采用确定性纤维追踪技术构建脑白质结构加权网络。比较三组间脑白质网络的节点连接强度差异,进一步采用受试者操作特征(receiver operator characteristic,ROC)曲线评估差异脑区对抑郁障碍和双相障碍鉴别诊断的价值。结果双相组在左前扣带回的节点强度较单相组降低(3.89±0.76 vs.4.74±0.60),在右尾状核(4.94±1.26 vs.3.46±0.99)、右苍白球(1.98±0.67 vs.1.25±0.29)的节点强度较单相组升高(P<0.01,FWE校正)。左前扣带回、右尾状核、右苍白球3个脑区的连接强度联合鉴别抑郁障碍和双相障碍绘制ROC曲线,曲线下面积(area under the curve,AUC)为0.95(95%CI:0.91~0.99;P<0.01),敏感度0.89,特异度0.87。结论脑结构网络的节点强度差异可以作为一个潜在的影像学生物标志物识别抑郁障碍和双相障碍,联合差异脑区的节点强度可以得到更好的识别率。
